If $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$

Question

I want to show that if $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$. It feels like I'm missing something really obvious. $X$ is homotopic to $Y$, so there exists $f: X \to Y$ and $g: Y \to X$ with $g \circ f \simeq 1_X$ and $f \circ g \simeq 1_Y$. Since $X$ is contractible, $1_X \simeq c$, where $c$ is a constant function. I want to show that $1_Y \simeq d$, where $d$ is some constant function. I can't use inverses in any way, since I don't know if $f$ and $g$ are invertible, so I have to somehow write down a homotopy that takes $1_Y$ to a constant. My instructor is very fond of just giving us definitions during lectures, with no examples of how to work with them, so I have no idea how to go about a problem like this. Any explanations will be appreciated.